Close Packing

Packing and Interstices

As shown here, identical atoms (or any round object, like coins or pool
balls) can be arranged in a hexagonal pattern where each is surrounded by six neighbors.
This is the tightest possible packing.

The left diagram shows what happens when we place a second layer (light gray) on top of
a lower layer (dark gray). Each upper atom nestles into a pocket formed by three other
atoms, enclosing a small space called an interstice. In this case, the four atoms
occupy the corners of a tetrahedron, so the openings are called tetrahedral
interstices. There are two sets. One is enclosed by an upper atom resting on three
below (shown in purple). Another consists of a cluster of three upper atoms resting on one
below (shown in blue). In both cases the centers of the four atoms are the vertices of a
tetrahedron (shown in yellow) but in one case the tetrahedron points up, in the other it
points down.

It turns out there are much larger openings between the sheets. A cluster of three
upper atoms can also rest on a cluster of three lower atoms (shown in red). These six
atoms lie at the corners of an octahedron, so these openings are called octahedral
interstices. In real minerals the close-packed sheets are often anions (usually
oxygen) with cations in the interstices. Some of the tetrahedral and octahedral
interstices are shown between the atoms (blue and yellow, respectively)

On the right is a representation of a pair of close-packed sheets just showing the
shapes of the interstices. Each vertex is the site of an atom. We have a sheet of
octahedra (blue top faces) separated by tetrahedra. The top faces of the tetrahedra are
omitted and only the downward-pointing tetrahedra are shown. The upward-pointing
tetrahedra are hidden; their top vertices are located where three octahedra join.

We can see each octahedron has six atoms, but each atom is shared with three octahedra,
so for each octahedron there are 6(1/3)=2 atoms. Each upward-pointing tetrahedron can be
pictured as sharing the three base atoms with three neighbors plus having the top atom to
itself, or 3(1/3)+1=2 atoms. The same is true for downward-pointing tetrahedra. Thus there
are equal numbers of octahedra, upward-pointing tetrahedra and downward-pointing
tetrahedra, and two atoms for each polyhedron. There are two tetrahedra for each
octahedron, one upward and one downward.

Hexagonal Close Packing

Top: Identical atoms can be packed into a sheet with a
hexagonal pattern. Another layer can be placed on top in one of two ways: over the
upward-pointing gaps (blue) or the downward-pointing gaps (yellow). Layers B and C show
layers in these positions.

Bottom: If layers alternate (left) or are
randomly stacked (right) the overall structure has only the hexagonal symmetry of the
individual sheets. The alternating pattern is called Hexagonal close packing. The
oxygen atoms in corundum and hematite have this packing.

In real crystals the anions often have close packing with the interstices between
layers occupied by cations. Since the cations attract the anion sheets and repel nearby
cations, electrical forces tend to produce symmetrical arrangements. Thus the regular
alternating stacking is favored.

Cubic Close Packing

Top: The three possible arrangements of close-packed
atomic layers are shown. Note how the three layers also form a square lattice of atoms.
The lattice plane is tilted in this view.

Bottom: If the layers repeat
cyclically, cubic close packing results. The two right diagrams show how
a cube can be formed by this packing. Oxygen atoms in spinel and magnetite have this
arrangement, as do the chlorine atoms in halite, the sulfur atoms in sphalerite and the
calcium atoms in fluorite.